Packing trees into 1-planar graphs

Felice De Luca, Emilio Di Giacomo, Seok Hee Hong, Stephen Kobourov, William Lenhart, Giuseppe Liotta, Henk Meijer, Alessandra Tappini, Stephen Wismath

Research output: Contribution to journalArticlepeer-review

Abstract

We introduce and study the 1-planar packing problem: Given k graphs with n vertices G1, …, Gk, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each Gi is a tree and k = 3. We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with n ≥ 12 vertices admits a 1-planar packing, while such a packing does not exist if n ≤ 10.

Original languageEnglish (US)
Pages (from-to)605-624
Number of pages20
JournalJournal of Graph Algorithms and Applications
Volume25
Issue number2
DOIs
StatePublished - 2021

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)
  • Geometry and Topology
  • Computer Science Applications
  • Computational Theory and Mathematics

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